Sharpness of the phase transition in percolation models

We consider a model of oriented percolation on Z d × Z, d > 2, with long-range interactions, in which the bond occupation probability decays as the α-stable distribution with α = 1. We use the lace expansion to get an L 1 infrared bound estimate which implies several critical exponents via the triangle condition.

Physical Review C, 1997

We examine the average cluster distribution as a function of lattice probability for a very small (Lϭ6) lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of Lϭ6 and Lϭ63 lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent and the ratio of critical exponents ␤/. ͓S0556-2813͑97͒02703-9͔

Physical Review E

We investigate percolation on a randomly directed lattice, an intermediate between standard percolation and directed percolation, focusing on the isotropic case in which bonds on opposite directions occur with the same probability. We derive exact results for the percolation threshold on planar lattices, and present a conjecture for the value the percolation-threshold for in any lattice. We also identify presumably universal critical exponents, including a fractal dimension, associated with the strongly-connected components both for planar and cubic lattices. These critical exponents are different from those associated either with standard percolation or with directed percolation.

Physical Review B

The Monte Carlo percolation-probability data of Frisch et al. are analyzed under the assumption that R(p), the probability that a given site (bond) belongs to an infinite cluster as a function of the probability p of site (bond) occupation, has the asymptotic behavior R(p) (p-p~)~, as p p~, where p~i s the critical percolation probability. The estimated values of P for various lattices are tabulated.

Physica A: Statistical Mechanics and its Applications

Can there be a 'Litmus test' for determining the nature of transition in models of percolation? In this paper we argue that the answer is in the affirmative. All one needs to do is to measure the 'growth exponent' χ of the largest component at the percolation threshold; χ < 1 or χ = 1 determines if the transition is continuous or discontinuous. We show that a related exponent η = 1 − χ which describes how the average maximal jump sizes in the Order Parameter decays on increasing the system size, is the single exponent that describes the finite-size scaling of a number of distributions related to the fastest growth of the Order Parameter in these problems. Excellent quality scaling analysis are presented for the two single peak distributions corresponding to the Order Parameters at the two ends of the maximal jump, the bimodal distribution constructed by interpolation of these distributions and for the distribution of the maximal jump in the Order Parameter.

Physical Review E, 2014

We derive the critical nearest-neighbor connectivity g n as 3/4, 3(7 − 9p tri c)/4(5 − 4p tri c), and 3(2 + 7p tri c)/ 4(5 − p tri c) for bond percolation on the square, honeycomb, and triangular lattice, respectively, where p tri c = 2 sin(π/18) is the percolation threshold for the triangular lattice, and confirm these values via Monte Carlo simulations. On the square lattice, we also numerically determine the critical next-nearest-neighbor connectivity as g nn = 0.687 500 0(2), which confirms a conjecture by Mitra and Nienhuis [J. Stat. Mech. (2004) P10006], implying the exact value g nn = 11/16. We also determine the connectivity on a free surface as g surf n = 0.625 000 1(13) and conjecture that this value is exactly equal to 5/8. In addition, we find that at criticality, the connectivities depend on the linear finite size L as ∼ L yt −d , and the associated specific-heat-like quantities C n and C nn scale as ∼ L 2yt −d ln(L/L 0), where d is the lattice dimensionality, y t = 1/ν the thermal renormalization exponent, and L 0 a nonuniversal constant. We provide an explanation of this logarithmic factor within the theoretical framework reported recently by Vasseur et al.

Journal of Statistical Physics, 1984

Various inequalities are derived and used for the study of the critical behavior in independent percolation models. In particular, we consider the critical exponent 7 associated with the expected cluster size X, and the structure of the n-site connection probabilities 9 = %(xl,..., xn). It is ...

Lack of self-averaging originates in many disordered models from a fragmentation of the phase space where the sizes of the fragments remain sampledependent in the thermodynamic limit. On the basis of new results in percolation theory, we give here an argument in favour of the conjecture that critical two dimensional percolation on the square lattice lacks of self-averaging.

2003

We analyze a deterministic cellular automaton σ · = (σn: n ≥ 0) corresponding to the zero-temperature case of Domany’s stochastic Ising ferromagnet on the hexagonal lattice H. The state space SH = {−1,+1} H consists of assignments of −1 or +1 to each site of H and the initial state σ0 = {σ0 x}x∈H is chosen randomly with P(σ0 x = +1) = p ∈ [0,1]. The sites of H are partitioned in two sets A and B so that all the neighbors of a site x in A belong to B and vice versa, and the discrete time dynamics is such that the σ · x’s with x ∈ A (respectively, B) are updated simultaneously at odd (resp., even) times, making σ · x agree with the majority of its three neighbors. In [1] it was proved that there is a percolation transition at p = 1/2 in the percolation models defined by σ n, for all times n ∈ [1, ∞]. In this paper, we study the nature of that transition and prove that the critical exponents β, ν and η of the dependent percolation models defined by σ n,n ∈ [1, ∞], have the same values ...

Physical Review B, 1990

Series expansions for general moments of the bond-percolation cluster-size distribution on hypercubic lattices to 15th order in the concentration have been obtained. This is one more than the previously published series for the mean cluster size in three dimensions and four terms more for higher moments and higher dimensions. Critical exponents, amplitude ratios, and thresholds have been calculated from these and other series by a variety of independent analysis techniques. A comprehensive summary of extant estimates for exponents, some universal amplitude ratios, and thresholds for percolation in all dimensions is given, and our results are shown to be in excellent agreement with the ε expansion and some of the most accurate simulation estimates. We obtain threshold values of 0.2488±0.0002 and 0.180 25±0.000 15 for the three-dimensional bond problem on the simple-cubic and body-centered-cubic lattices, respectively, and 0.160 05±0.000 15 and 0.118 19±0.000 04, for the hypercubic bond problem in four and five dimensions, respectively. Our direct exponent estimates are γ=1.805±0.02, 1.435±0.015, and 1.185±0.005, and β=0.405±0.025, 0.639±0.020, and 0.835±0.005 in three, four, and five dimensions, respectively.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1981

We prove that the critical probabilities of site percolation on the square lattice satisfy the relation pc+p*=l. Furthermore we prove the continuity of the function "percolation probability".

Journal of Physics A: Mathematical and General, 1996

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent β are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: q c = 0.4043528 ± 0.0000010 and β = 0.27645 ± 0.00010; in the bond case: q c = 0.52198 ± 0.00001 and β = 0.2769 ± 0.0010; and in the site-bond case: q c = 0.264173 ± 0.000003 and β = 0.2766 ± 0.0003. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent ∆ = 1.

arXiv (Cornell University), 2020

We develop techniques to study the phase transition for planar Gaussian percolation models that are not (necessarily) positively correlated. These models lack the property of positive associations (also known as the 'FKG inequality'), and hence many classical arguments in percolation theory do not apply. More precisely, we consider a smooth stationary centred planar Gaussian field f and, given a level ∈ R, we study the connectivity properties of the excursion set {f ≥ − }. We prove the existence of a phase transition at the critical level crit = 0 under only symmetry and (very mild) correlation decay assumptions, which are satisfied by the random plane wave for instance. As a consequence, all non-zero level lines are bounded almost surely, although our result does not settle the boundedness of zero level lines ('no percolation at criticality'). To show our main result: (i) we prove a general sharp threshold criterion, inspired by works of Chatterjee, that states that 'sharp thresholds are equivalent to the delocalisation of the threshold location'; (ii) we prove threshold delocalisation for crossing events at large scales-at this step we obtain a sharp threshold result but without being able to locate the threshold-and (iii) to identify the threshold, we adapt Tassion's RSW theory replacing the FKG inequality by a sprinkling procedure. Although some arguments are specific to the Gaussian setting, many steps are very general and we hope that our techniques may be adapted to analyse other models without FKG. Contents 1. Introduction 2 2. Proof of the main results 10 3. Sharp thresholds are equivalent to the delocalisation of the threshold location 20 4. Delocalisation of the threshold location 24 Appendix A. Basic properties of smooth Gaussian fields 30 Appendix B. Stratified Morse functions 31 Appendix C. RSW theory (by Laurin Köhler-Schindler) 33 References 39

Journal of Aerosol Science, 1977

For a lattice gas with attractive potentials of finite range we use the inequalities of Fortuin, Kasteleyn, and Ginibre (FKG) to obtain fairly accurate upper and lower bounds on the equilibrium probability p(K) of finding the set of sites K occupied and the adjacent sites unoccupied, i.e., on the probabilities of finding specified clusters. The probability that a given site, say the origin, is empty or belongs to a cluster of at most l particles is shown to be a nonincreasing function of the fugacity z and the reciprocal temperature fl = (kT)-1; hence the percolation probability is a nondecreasing function of z and ft. If the forces are not entirely attractive, or if the ensemble is restricted by forbidding clusters larger than a certain size, the FKG inequalities no longer apply, but useful upper and lower bounds on p(K) can still be obtained if the density of the system and the size of the cluster K are not too large. They are obtained from a generalization of the Kirkwoo~Salsburg equation, derived by regarding the system as a mixture of different types of cluster, whose only interaction is that they cannot overlap or touch.